Statistical inference for a constant-stress partially accelerated life tests based on progressively hybrid censored samples from inverted Kumaraswamy distribution

In this article, we investigate the problem of point and interval estimations under constant-stress partially accelerated life tests. The lifetime of items under use condition is assumed to follow the two-parameter inverted Kumaraswamy distribution. Based on Type-I progressively hybrid censored samples, the maximum likelihood and Bayesian methods are applied to estimate the model parameters as well as the acceleration factor. Under linear exponential, general entropy and squared error loss functions, Bayesian method outcomes are obtained. In addition, interval estimation is achieved by finding approximately confidence intervals for the parameters, as well as credible intervals. To investigate the accuracy of the obtained estimates and to compare the performance of confidence intervals, a Monte Carlo simulation is developed. Finally, a set of real data is analyzed to demonstrate the estimation procedures.


Introduction
In some situations, due to the continuous progress in manufacturing technology, modern products are designed and built to operate without failure for a long time under standard operating conditions, and then it is very difficult, if not impossible, to obtain failure data for these products. Because of this difficulty, reliability professionals have attempted to devise techniques to quickly induce failures by subjecting the highly reliable products to harsher environmental conditions without adding additional failure modes other than those found under standard operating conditions. To evaluate the life characteristics and reliability performance of such products under standard operating conditions, accelerated life tests (ALTs) are used to quickly obtain information on such products by subjecting them to severe conditions to obtain the required failure data in a short time. The severe conditions may include subjecting the test products to higher voltage, temperature, pressure, vibrations, or combining some of them. 1 À ½1 À ð1 þ xÞ À g � y : The PDF and HRF curves of IKum distribution show that it has a long right tail when compared to other commonly used distributions, see Fig 1. As a result, it will have an impact on long-term reliability predictions, producing optimistic predictions of rare events that occur in the right tail of the distribution when compared to other distributions. In addition, the IKum distribution fits several data sets in the literature well.
The IKum distribution was proposed by Abd AL-Fattah et al. [21]. They investigated various structural properties with applications. They also addressed the estimation problem of parameters of the IKum distribution based on Type-II censoring. AL-Dayian et al. [22] estimated the shape parameters, reliability function (RF) and HRF of IKum based on dual generalized order statistics. Bagci et al. [23] employed ML, least squares, and maximum product of spacing methodologies in estimating the parameters of the IKum distribution.
Types-I and II CSs are frequently used by the experimenters. A hybrid CS (HCS) is considered a mixture of them. It was introduced by Epstin [24]. One of the drawbacks of the conventional Type-I, Type-II, or HCSs is that they do not permit removal of items at points other than the experiment terminal point. One CS known as the progressive Type-II CS, which permits the experimenter to remove items at several points other than the experiment terminal point, has become very popular in the last few years. Kundu and Joarder [25] proposed a new CS named Type-I progressive HCS (Type-I PHCS) and studied the ML estimates (MLEs) and Bayes estimates for an exponential distribution. The progressive HCS can provide more information about the tail of the distribution under consideration. Furthermore, it can overcome the drawbacks associated with Type-I and Type-II CSs. For example, in Type-II censoring, the experiment termination time is uncontrolled, whereas the efficiency level is controlled in Type-I censoring. For more details and some recent references on CSs, see [26][27][28][29][30][31][32][33][34][35][36].
Suppose that n identical items are placed on a life test. The procedure of progressive Type-II censoring assumes that the integers censoring values R 1 , R 2 , . . ., R m , 1 � m � n, are assigned in advance with R j � 0 and P m j¼1 R j þ m ¼ n. Let T be a prescribed time point. The Type-I PHCS involves the termination of the life test at the time T � = min(X m:m:n , T). If the m-th failure occurs before T, then the experiment is terminated at the time point X m:m:n . Otherwise, the experiment stops at time T satisfying X k:m:n � T < X k+ 1:m:n , and all the remaining R � k ¼ n À R 1 À . . . À R k À k live test items are removed, see Fig 2. Here, k represents the number of failures occurring up to the time point T. We denote these two cases as Case I and Case II, respectively.
Case I: X 1:m:n < X 2:m:n < . . . < X m:m:n , if X m:m:n � T. Case II: X 1:m:n < X 2:m:n < . . . < X k:m:n , if T < X m:m:n , X k:m:n � T < X k + 1:m:n . The novelty of this article is to apply the constant-stress PALT to items whose lifetimes under normal stress conditions follow the IKum distribution under Type-I PHCS and to estimate the involved parameters using ML and Bayes (under linear exponential (LINEX), general entropy (GE), and squared error (SE) loss functions) methods. A real data set is analyzed to demonstrate and assess the performance of the introduced estimation methods.
The remaining sections of the article are arranged as follows: In Section 2, the model description is provided.MLEs and asymptotic confidence intervals (CIs) of the unknown parameters are discussed in Section 3.In Section 4, using the Markov chain Monte Carlo (MCMC) method, Bayes estimates, and credible intervals for the parameters are obtained. A simulation study followed by results is presented in Section 5. In Section 6, a real data set is analyzed to demonstrate and assess the performance of the introduced estimation methods given in Sections 3 and 4. Finally, Section 7 is devoted to some concluding remarks.

Model description
In constant-stress PALT, the total test items are divided into two groups. Let n 1 be the number of items of group 1 that fail at standard stress s 0 with total lifetime Y = T. The CDF and PDF of an item's total lifetime Y in group 1 are then given, respectively, by y > 0 ; ðg; y > 0Þ; The life testing experiment is finished at time T 1 when T 1 is reached before occurring the m 1 -th failure. Otherwise, the experiment is finished as soon as the m 1 -th failure occurs, i.e., the end time is a random variable T � 1 ¼ minfY 1 m 1 :m 1 :n 1 ; T 1 g: Let n 2 be the number of items of group 2 that fail at accelerated stress s 1 with total lifetime Y = λ −1 T, where T is the lifetime of an item at standard stress and λ is an acceleration factor (λ > 1). The CDF and PDF of an item's total lifetime Y in group 2 are then given, respectively, by The life testing experiment is finished at time T 2 when T 2 is achieved before occurring the m 2th failure. Otherwise, the experiment is finished as soon as the m 2 -th failure occurs, i.e., the end time is a random variable Therefore, under Type-I PHCS, for j = 1, 2, we have the next two cases: Case I:

Maximum likelihood estimation
In this section, we construct the likelihood function (LF), based on observed data that are subject to Type-I PHCS, to obtain ML estimators of the parameters γ, θ, and λ. From CDF (1) and its corresponding PDF (2), we obtain the LFs of γ, θ and λ.
The ML estimators can be got by solving the nonlinear equations given in (9) with respect to (γ, θ, λ). It is clear that system of nonlinear equations (9) is not in an analytically tractable form. Therefore, a suitable numerical method must be applied to get MLEs of γ, θ, and λ. An iterative algorithm such as the Newton-Raphson can be utilized to get the MLEsĝ,ŷ andl.

Asymptotic confidence interval
In this subsection, the approximate CIs of the unknown parameters (γ, θ, λ) are deduced based on the asymptotic distribution of the MLEs.
The NACIs obtained above may occasionally have negative lower bounds. To overcome this problem, Meeker and Escobar [38], and Xu and Long [39] considered approximate CI for the parameters using the log-transformation and delta approaches. Two-sided 100(1 − �)% normal approximation, based on log-transformed ML estimators, CIs (LTCIs) for γ, θ and λ are then given by �ĝ exp ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi varðĝÞ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi varðĝÞ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi varðŷÞ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi varðŷÞ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi varðlÞ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi varðlÞ q � =l i� :

Bayes estimation
In Bayes analysis, MCMC method is a general computing method applicable to both noninformative and informative priors. It makes Bayesian analysis attractive for a complex model. In the current section, we discuss the Bayes estimators of the parameters under the assumptions of informative and non-informative priors for γ, θ and λ. Bayes estimators are considered under LINEX, GE, and SE loss functions as we present below.
Since the invariance property does not hold with the Bayes estimation method, then the Bayes estimators of the parameters γ, θ and λ, in addition to the RF and HRF should be calculated separately.
For simplicity, assume that G = G(α) � G(γ, θ, λ) is a function of vector of parameters to be estimated. Clearly, G may assume one of the parameters or a combination of them (such as G� RF or G� HRF). Then the BE of G based on LINEX, GE, and SE loss functions are given by, see [40][41][42], Assume that the experimenter's prior belief is described by a function π(γ, θ, λ) and that the parameters γ, θ and λ are independent random variables. Let π 1 (γ) and π 2 (θ), respectively, denote the prior densities for γ and θ, following gamma distributions and let π 3 (λ) denote a non-informative prior for λ, with respective densities given by, p 1 ðgÞ / g mÀ 1 e À g=n ; ðg; m; nÞ > 0; p 2 ðyÞ / y zÀ 1 e À y=Z ; ðy; z; ZÞ > 0; Therefore, the joint prior distribution of γ, θ and λ can be written as pðg; y; lÞ / g mÀ 1 y zÀ 1 l À 1 e À ðg=nþy=ZÞ : The informative gamma prior distribution is used in this article because it is more appropriate than non-informative prior. Merging LF (7) with joint prior distribution of γ, θ and λ (11), we get the joint posterior density of γ, θ and λ, given the data, as follows: From (12), it is obvious that the Bayes estimators of γ, θ and λ cannot be got in closed forms. Therefore, we adopt MCMC method to obtain them.

Markov chain Monte Carlo
MCMC has an important role in Bayesian inference.The Gibbs sampling and Metropolis-Hastings algorithms are the two most widely used MCMC methods that are applied in statistics, statistical physics, signal processing, digital communications, machine learning, etc. Using joint posterior density function (12), the conditional posterior distributions of model parameters γ, θ, λ can be written, respectively, in the forms ð1 þ ly 2i:m 2 :n 2 Þ À ðgþ1Þ ½1 À ð1 þ ly 2i:m 2 : The conditional posterior PDFs of γ, θ, and λ do not belong to well-known distributions, and hence generating samples from (13) is not available, directly. Consequently, with normal proposal distribution, Metropolis-Hastings algorithm is applied to get Bayesian estimates for γ, θ, and λ.
The following MCMC procedure is suggested to evaluate the Bayes estimates of γ, θ, and λ and the corresponding credible intervals.
• Step 8. Get the Bayes estimates of γ, θ, and λ based on SE loss function aŝ where M is the burn-in period.
• Step 9. Get the Bayes estimates of γ, θ, and λ based on LINEX loss function aŝ Step 10. Get the Bayes estimates of γ, θ, and λ based on GE loss function aŝ Step 11: To calculate the credible intervals for γ, order γ (M+ 1) , . . ., γ (N) as γ [1] Then the 100(1 − �)% credible interval for γ is given by [w] denotes the greatest integer less than or equal to w Similarly, it is also possible to construct credible intervals for θ and λ.

Simulation study
We are performing a simulation analysis in this section to compare the performance of various estimates and CIs. The different estimates (MLE and Bayes estimates) are compared in terms of mean squared errors (MSEs) and relative absolute biases (RABs). Also, the different intervals (asymptotic and credible intervals) are compared in terms of average lengths and coverage probabilities. The simulation study is done according to the following procedure.
• Step 3: Generate two random samples from CDFs F 1 (y) and F 2 (y), given in (1) and (2), respectively, and then apply the progressive hybrid censoring technique to them to generate the two samples y j1 ; . . . ; y jD j , j = 1, 2.  Table 1 presents the various CSs, applied in the simulation study, for various choices of sample sizes n j , j = 1, 2, and observed failure times m j which describe 60% and 80% of the sample size. In Tables 2, 3, 5 and 6, we provide the MSEs and RABs of ML and Bayes estimates for the parameters γ, θ and λ relative to SE, LINEX and GE (with different values of c) loss functions, considering T 1 = 0.9, and T 2 = 0.5. The average interval lengths (AILs) and the corresponding 95% coverage probabilities (CPs), using the asymptotic distributions of MLEs, and credible intervals are displayed in Tables 4 and 7. The prior parameters have been chosen to be μ = 2, v = 1.5, z = 1.5 η = 1.5, which yield the generated values γ = 1.85699 and θ = 2.50899 (as true values) with λ = 1.5, considering some different Type-I PHCSs as in Table 1 with notation that (3 � 0, 1) means (0, 0, 0, 1).

Simulation results
Based on the numerical calculations presented in Tables 2-7, we can observe the following points: 1. The MSEs and RABs decrease as the value of sample size increases for all cases. 3. The results presented in Tables 2 and 3 are based on different sample sizes (n 1 < n 2 ) while those presented in Tables 5 and 6 are based on the same sample sizes (n 1 = n 2 ). Better results are obtained when considering n 1 < n 2 . This can be seen when we compare the results presented in Tables 2 and 3 with those in Tables 5 and 6. This is due to a greater number of units subject to acceleration will have smaller lifetimes (and hence smaller MSEs) than those under normal stress conditions. 6. For fixed values of the sample size, by increasing the observed failure times, the AILs for the credible intervals are smaller than those for the NACIs and LTCIs. Also, the CPs of the credible intervals are closer to 95% than those for NACIs and LTCIs.
7. The AILs of the NACIs, LTCIs and credible intervals decrease as the sample size increases.

Real data analysis
In this section, a real data set is introduced to show how the ML and Bayes estimation methods work in practice based on real data from Nelson [1]. The data are presented in Table 8. They correspond to the oil breakdown times of insulating fluid under two stress levels (34 kilovolt (kv) and 36 kv), considering the data set under 34 kv as data under normal stress. Before  further proceeding, we test the validity of IKum distribution to fit the data listed in Table 8 using Kolmogorov-Smirnov (K-S) test statistic and its corresponding p-value for each stress level. The results are listed in Table 9 in which we can notice that the IKum distribution fits the given data, under the two stress levels, well because the p-values are greater than 0.05. This is done graphically by plotting the empirical CDFs of the two data sets against the CDFs of the IKum distribution, see the two panels in   We apply Type-I PHCS to the real data set listed in Table 8 to obtain three samples subjecting to Type-I PHCS see Table 10, where the three Type-I PHCSs are considered as follows: We assume that n 1 = 19, m 1 = 15, T 1 = 34, n 2 = 15, m 2 = 12, T 2 = 12 and CS I: R 1i = (4, 14 � 0) and R 2i = (3, 11 � 0). CS II: R 1i = (14 � 0, 4) and R 2i = (11 � 0, 3).
CS III: R 1i = (1, 1, 1, 1, 11 � 0) and R 2i = (1, 1, 1, 9 � 0).   Based on the three samples shown in Table 10, which are subject to Type-I PHCS, the ML and Bayes estimates of the parameters γ, θ, λ are calculated and presented in Table 11, while the 95% confidence and credible intervals are presented in Table 12. In addition, the mean time to failure (MTTF) and RF (R 1 ðyÞ ¼ 1 À F 1 ðyÞ) for certain mission times are calculated and given in Table 11 under normal operating conditions.

Conclusion
When the lifetimes of items under use conditions are subject to the IKum distribution, we have addressed the issue of point and interval estimations under constant-stress PALT. To estimate the model parameters, the ML and Bayes (under LINEX, GE, and SE loss functions) methods have been applied based on Type-I PHCS. In addition, the estimated CIs have been   acquired as well as credible intervals. To get Bayes estimates of the model parameters, the MCMC technique has been implemented. To investigate the accuracy of the estimates obtained and to compare the output of the CIs, a simulation study has been developed. To discuss and test the efficiency of the suggested estimation methods, a real data set has been considered. The actual data results show that the IKum distribution is a good candidate for fitting the data and the methods of estimation perform well under Type-I PHCS as well. Finally, we It should be noted that if the hyperparameters are unknown, we can use the empirical Bayes method to estimate them using past samples, see [43]. Instead, one may use the hierarchical Bayes approach in which suitable priors for the hyperparameters could be used, see [44].
Future work: Neutrosophic statistics is the extension of classical statistics and is applied when the data are coming from a complex process or an uncertain environment. So, the current study can be extended using neutrosophic statistics as future research, see [45][46][47].   Table 8, under two the stress levels.